Method for determining contribution to a fingerprint

ABSTRACT

A method, system and program for determining a fingerprint of a parameter. The method includes determining a contribution from a device out of a plurality of devices to a fingerprint of a parameter. The method includes obtaining parameter data and usage data, wherein the parameter data is based on measurements for multiple substrates having been processed by the plurality of devices, and the usage data indicates which of the devices out of the plurality of the devices were used in the processing of each substrate; and determining the contribution using the usage data and parameter data.

This application is a continuation of U.S. patent application Ser. No.17/023,474, filed Sep. 17, 2020, now allowed, which is a continuation ofU.S. patent application Ser. No. 16/461,044, filed May 15, 2019, nowU.S. Pat. No. 10,816,904, which is a U.S. national phase entry of PCTpatent application no. PCT/EP2018/063527, filed May 23, 2018, whichclaims the benefit of priority of U.S. patent application No.62/523,531, filed Jun. 22, 2017, and of U.S. patent application No.62/639,481, filed Mar. 6, 2018, each of the foregoing applications isincorporated herein in its entirety by reference.

FIELD

The present description relates to a method, system and program fordetermining contribution to a fingerprint of a parameter.

BACKGROUND

A lithographic apparatus is a machine that applies a desired patternonto a substrate, usually onto a target portion of the substrate. Alithographic apparatus can be used, for example, in the manufacture ofintegrated circuits (ICs). In that instance, a patterning device, whichis alternatively referred to as a mask or a reticle, may be used togenerate a circuit pattern to be formed on an individual layer of theIC. This pattern can be transferred onto a target portion (e.g.,including part of, one, or several dies) on a substrate (e.g., a siliconwafer). Transfer of the pattern is typically via imaging onto a layer ofradiation-sensitive material (resist) provided on the substrate. Ingeneral, a single substrate will contain a network of adjacent targetportions that are successively patterned. These target portions arecommonly referred to as “fields”.

In lithographic processes, it is desirable to frequently measure thestructures created forming a circuit pattern, e.g., for process controland verification. Various tools for making such measurements are known,including scanning electron microscopes, which are often used to measurecritical dimension (CD), and specialized tools to measure overlay (whichis the accuracy of alignment of two layers in an at least partiallypatterned substrate). Various techniques can be used to measureperformance of the lithographic process. This in turn allowssophisticated process corrections to be included in the control of theoperations performed by the lithographic apparatus.

SUMMARY

After exposure of the substrate, i.e. when the patent is transferredunder the substrate, various different processes can be carried out onthe substrate. Thus, different processing tools may be used on asubstrate. However, each processing tool may slightly vary the overlayfingerprint. There are methods of reducing the effects of variationsinduced by processing tools, for example by adjusting lithographicapparatus actuators and improving the control. However, each substratemay be provided with the same correction meaning that the variationinduced on each substrate is not individually dealt with.

One issue is that a number of different devices may be provided to carryout the processing of a substrate. There may be a number of differenttypes of device used to process a substrate in different ways, e.g. anetching device and a polishing device. Additionally, there may be amultiple of the same type of device available to be used for eachsubstrate. Thus, there may be multiple etching and/or polishing deviceswhich could be used on each substrate and each substrate may beprocessed using a different combination of devices from a plurality ofdevices. Each of the devices may have a different effect on thefingerprint of a parameter and it is not necessarily clear what theeffect of an individual device is on the parameter of the substrate.

An embodiment of the present invention has, for example, an aim ofdetermining a contribution from a device out of a plurality of devicesto a fingerprint of a parameter, the parameter being associated withprocessing of a substrate.

According to an aspect, there is provided a method for determining acontribution of a device out of a plurality of devices to a fingerprintof a parameter, the parameter being associated with processing of asubstrate, the method comprising: obtaining parameter data and usagedata, wherein the parameter data is based on measurements for multiplesubstrates having been processed by the plurality of devices, and theusage data indicates which of the devices out of the plurality of thedevices were used in the processing of each substrate; and determiningthe contribution using the usage data and parameter data.

According to an aspect, there is provided a system comprising aprocessor configured to determine a contribution of a device out of aplurality of devices to a fingerprint of a parameter, the parameterbeing associated with processing of a substrate, the processor beingconfigured to: obtain parameter data and usage data, wherein theparameter data is based on measurements for multiple substrates havingbeen processed by the plurality of devices, and the usage data indicateswhich of the devices out of the plurality of devices were used in theprocessing of each substrate; and determine the contribution using theusage data and the parameter data.

According to an aspect, there is provided a program for controllingdetermining a contribution of a device out of a plurality of devices toa fingerprint of a parameter, the parameter being associated withprocessing of a substrate, the program comprising instructions forcarrying out a method comprising: obtaining parameter data and usagedata, wherein the parameter data is based on measurements for multiplesubstrates having been processed by the plurality of devices, and theusage data indicates which of the devices out of the plurality ofdevices were used in the processing of each substrate; and determiningthe contribution using the usage data and the parameter data.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way of exampleonly, with reference to the accompanying schematic drawings in which:

FIG. 1 depicts a lithographic apparatus together with other apparatusesforming a production facility for devices, as an example of a system inwhich an embodiment of the invention may be used; and

FIG. 2 depicts the lithographic apparatus of FIG. 1 in which a pluralityof devices are illustrated by way of example only.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

Before describing embodiments of the invention in detail, it isinstructive to present an example environment in which embodiments ofthe present invention may be implemented. The invention can be applied,for example, after at least partial processing of a substrate in alithographic manufacturing process. The invention can be applied forexample to determine the effect of a device which is part of alithographic apparatus. A lithographic process for the manufacture ofsemiconductor devices will be described to provide an exemplary contextin which the method can be used. The principles of the presentdisclosure can be applied in other processes without limitation.

FIG. 1 shows a lithographic apparatus LA at 100 as part of an industrialfacility implementing a high-volume, lithographic manufacturing process.In the present example, the manufacturing process is adapted for themanufacture of semiconductor products (e.g., integrated circuits) onsubstrates such as semiconductor wafers. The skilled person willappreciate that a wide variety of products can be manufactured byprocessing different types of substrates in variants of this process.The production of semiconductor products is used purely as an examplewhich has great commercial significance today.

Within the lithographic apparatus (or “litho tool” 100 for short), ameasurement station MEA is shown at 102 and an exposure station EXP isshown at 104. A control unit LACU is shown at 106. In this example, eachsubstrate visits the measurement station and the exposure station tohave a pattern applied. In an optical lithographic apparatus, forexample, a projection system is used to transfer a product pattern froma patterning device MA onto the substrate using conditioned radiationand a projection system. This is done by forming an image of the patternin a layer of radiation-sensitive resist material on a substrate.

The term “projection system” used herein should be broadly interpretedas encompassing any type of projection system, including refractive,reflective, catadioptric, magnetic, electromagnetic and electrostaticoptical systems, or any combination thereof, as appropriate for theexposure radiation being used, and/or for other factors such as the useof an immersion liquid or the use of a vacuum. The patterning device MAmay be a mask or reticle, which imparts a pattern to a radiation beamtransmitted or reflected by the patterning device MA. Well-known modesof operation include a stepping mode and a scanning mode. The projectionsystem may cooperate with support and positioning systems for thesubstrate and the patterning device in a variety of ways to apply adesired pattern to many target portions across a substrate. Programmablepatterning devices may be used instead of reticles having a fixedpattern. The radiation for example may include electromagnetic radiationin the deep ultraviolet (DUV) or extreme ultraviolet (EUV) wavebands.The present disclosure is also applicable to other types of lithographicprocess, for example imprint lithography and direct writing lithography,for example by electron beam.

The lithographic apparatus control unit LACU controls all the movementsand measurements of various actuators and sensors, causing the apparatusto receive substrates W and patterning devices MA and to implement thepatterning operations. Control unit LACU also includes signal processingand data processing capacity to implement desired calculations relevantto the operation of the apparatus. In practice, control unit LACU willbe realized as a system of many sub-units, each handling the (real-time)data acquisition, processing and control of a subsystem or componentwithin the lithographic apparatus LA.

In an embodiment, before the pattern is applied to a substrate at theexposure station EXP, the substrate is processed at the measurementstation MEA so that various preparatory steps may be carried out. Thepreparatory steps may include mapping the surface height of thesubstrate using a level sensor and/or measuring the position ofalignment marks on the substrate using an alignment sensor. Thealignment marks are arranged nominally in a regular grid pattern.However, due to inaccuracies in creating the marks and also due todeformations of the substrate that occur throughout its processing, thealignment marks may deviate from the ideal grid. Consequently, inaddition to measuring position and orientation of the substrate, thealignment sensor in practice must measure in detail the positions ofmany marks across the substrate area, if the apparatus is to printproduct features at the correct locations with high accuracy.

The lithographic apparatus LA may be of a so-called dual stage typewhich has two substrate tables, each with a positioning systemcontrolled by the control unit LACU. While one substrate on onesubstrate table is being exposed at the exposure station EXP, anothersubstrate can be loaded onto the other substrate table at themeasurement station MEA so that various preparatory steps may be carriedout. The measurement of alignment marks is time-consuming and theprovision of two substrate tables enables a substantial increase in thethroughput of the apparatus. If the position sensor is not capable ofmeasuring the position of the substrate table while it is at themeasurement station as well as at the exposure station, a secondposition sensor may be provided to enable the positions of the substratetable to be tracked at both stations. Alternatively, the measurementstation and exposure station can be combined. For example, it is knownto have a single substrate table, to which a measurement stage istemporarily coupled during the pre-exposure measuring phase. The presentdisclosure is not limited to either type of system.

Within the production facility, apparatus 100, LA forms part of a “lithocell” or “litho cluster” that contains also a coating device 108, COATconfigured to apply photosensitive resist and/or other coatings tosubstrates W for patterning by the apparatus 100. At an output side ofapparatus 100, a baking device 110, BAKE and developing device 112, DEVare provided for developing the exposed pattern into a physical resistpattern. Between all of these devices, substrate handling systems takecare of supporting the substrates and transferring them from one pieceof apparatus/device to the next. These devices, which are oftencollectively referred to as the “track”, may be under the control of atrack control unit which is itself controlled by a supervisory controlsystem SCS, which may also control the lithographic apparatus viacontrol unit LACU. Thus, the different devices described above, anddevices which may be part of the lithographic apparatus, e.g. thesubstrate table, can be operated to maximize throughput and processingefficiency. Supervisory control system SCS receives recipe information Rwhich provides in great detail a definition of the steps to be performedto create each patterned substrate.

Once the pattern has been applied and developed in the litho cell,patterned substrates 120 are transferred to other processing devicessuch as are illustrated at 122, 124, 126. A wide range of processingsteps can be implemented by various devices in a typical manufacturingfacility. For the sake of example, device 122 in this example is anetching station ETCH, and device 124 is an apparatus ANNEAL thatperforms a post-etch annealing step. Further physical and/or chemicalprocessing steps are applied in further devices, 126, etc. Thesespecific devices are provided for example only, and other combinationsof devices could be used within the scope of the embodiments as aredescribed in further detail below. Additionally, the order in which asubstrate is processed by the devices is for example only and thedevices may be used in a different order not shown. Numerous types ofoperation can be carried out by such devices, such as deposition ofmaterial, modification of surface material characteristics (oxidation,doping, ion implantation, etc.), chemical-mechanical polishing (CMP),and so forth. The device 126 may, in practice, represent a series ofdifferent processing steps performed in one or more devices. Thesedevices may otherwise be referred to as apparatus.

The manufacture of semiconductor devices involves many repetitions ofsuch processing, to build up device structures with appropriatematerials and patterns, layer-by-layer on the substrate. Accordingly,substrates 130 arriving at the litho cluster may be newly preparedsubstrates, or they may be substrates that have been processedpreviously in this cluster or in another apparatus entirely. Similarly,depending on the desired processing, substrates 132 on leaving device126 may be returned for a subsequent patterning operation in the samelitho cluster, they may be destined for patterning operations in adifferent cluster, or the substrates 134 may be finished products to besent for dicing and packaging.

Each layer of the product structure typically involves a different setof process steps, and the devices 122, 124 and 126 used at each layermay be completely different in type. Further, even where the processingsteps to be applied by the devices 122, 124 and 126 are nominally thesame, in a large facility, there may be several supposedly identicalmachines working in parallel to perform the steps 122, 124 and 126 ondifferent substrates. In other words, there may be several devices ofthe same type, for example, several etching devices 122, severalannealing devices 124 and/or several of each type of further device 126.Small differences in set-up or faults between these machines can meanthat devices of the same type influence different substrates indifferent ways. Even steps that are relatively common to each layer,such as etching (device 122) may be implemented by several etchingapparatuses that are nominally identical but working in parallel tomaximize throughput. In practice, moreover, different layers typicallyinvolve different etch processes, for example chemical etches, plasmaetches, according to the details of the material to be etched, andspecial requirements such as, for example, anisotropic etching.

The previous and/or subsequent processes may be performed in otherlithography apparatuses, as just mentioned, and may even be performed indifferent types of lithography apparatus. For example, some layers inthe device manufacturing process which are very demanding in terms ofparameters such as resolution and/or overlay may be performed in a moreadvanced lithography tool than other layers that are less demanding.Therefore some layers may be exposed in an immersion type lithographytool, while others are exposed in a ‘dry’ tool. Some layers may beexposed in a litho tool working at DUV wavelengths, while others areexposed using EUV wavelength radiation.

In order that the substrates that are exposed by the lithographicapparatus are exposed correctly and consistently, it is desirable toinspect exposed substrates to measure one or more properties such asoverlay error between subsequent layers, a line thickness, a criticaldimension (CD), etc. Accordingly a manufacturing facility in which lithocell is located also includes a metrology system 140, MET which receivessome or all of the substrates W that have been processed in the lithocell. Metrology results are provided directly or indirectly to thesupervisory control system (SCS) 138. If an error is detected, one ormore adjustments may be made to one or more exposures of one or moresubsequent substrates, especially if the metrology can be done soon andfast enough that one or more other substrates of the same lot are stillto be exposed. Also, one or more already exposed substrates may bestripped and reworked to improve yield, or discarded, thereby avoidingperforming further processing on a substrate known to be faulty. In acase where only one or more target portions of a substrate are faulty,further exposures can be performed only on the one or more targetportions which are good.

As shown in FIG. 1, a metrology apparatus 140 can be provided for makingmeasurements of parameters of the products at desired stages in themanufacturing process. A common example of a metrology apparatus in amodern lithographic production facility is a scatterometer, for examplean angle-resolved scatterometer or a spectroscopic scatterometer, and itmay be applied to measure properties of the developed substrates at 120prior to etching in the apparatus 122. Using metrology apparatus 140, itmay be determined, for example, that important performance parameterssuch as overlay or critical dimension (CD) do not meet specifiedaccuracy requirements in the developed resist. Prior to the etchingstep, the opportunity exists to strip the developed resist and reprocessthe substrates 120 through the litho cluster. Additionally oralternatively, the metrology results 142 from the metrology apparatus140 can be used to maintain accurate performance of the patterningoperations in the litho cluster, by supervisory control system SCSand/or control unit LACU 106 making small adjustments over time, therebyreducing or minimizing the risk of products being madeout-of-specification, and requiring re-work. Of course, metrologyapparatus 140 and/or other metrology apparatuses (not shown) can beapplied to measure properties of the processed substrates 132, 134, andincoming substrates 130.

In the example of a lithographic manufacturing process, the substratesare semiconductor wafers or other substrates to which patterns are to beapplied in a patterning step, and structures formed by physical andchemical process steps.

An embodiment of the present invention provides a method for determininga contribution of a device out of a plurality of devices to afingerprint of a parameter associated with processing of a substrate.Thus, the contribution may be considered as a partial contribution or arelative contribution. This embodiment will be described with respect tothe devices shown in FIG. 2. As will be clear from FIG. 2, three etchingdevices are provided, 122A, 122B and 122C and three annealing devicesare provided 124A, 124B and 124C. The substrate 120 having beenprocessed in the lithographic apparatus, is then further processed in anetching step and annealing step. Other steps may also be included, forexample a developing step as shown in FIG. 2.

For etching the substrate 120, the substrate 120 may pass through afirst etching device 122A, a second etching device 122B or a thirdetching device 122C. For annealing the substrate 120, the substrate 120may pass through a first annealing device 124A, a second annealingdevice 124B or a third annealing device 124C. Each of the differentetching devices and annealing devices may have different influences onthe fingerprint of the resulting substrate 134. The fingerprint may be aspatial fingerprint, and may be an intra-field and/or inter-fieldfingerprint, such as a substrate fingerprint, or a field fingerprint, aslit fingerprint or anything time or sequence wise.

As indicated above, the different devices and apparatus used caninfluence different substrates in different ways. When a substrate hasbeen fully processed, such as substrate 134, even when measurements havebeen made of the parameters of the substrate, it is not necessarilyclear how each of the different devices, e.g. one or more of thedifferent etching devices 122A, 122B or 122C and/or one or more of thedifferent annealing devices 124A, 124B or 124C, have affected thefingerprint of a parameter of the substrate. As described above, evenwhen a plurality of different devices of the same type is used, they canaffect a substrate in different ways. It is desirable to produceconsistent substrates irrespective of the specific device used. Thus, itis desirable to determine the influence of the different devices on thesubstrate. It is beneficial to know how a device may affect a parameteron a substrate, for example, for diagnostic reasons or to improvecontrol. This information may be used in different ways, for example todecide when a specific device needs to have alterations made to it orcould be used as part of a feedback loop to control the processing of asubstrate as is described in further detail below.

So, in an embodiment, there is provided a method for determining acontribution of a device out of a plurality of devices to a fingerprintof a parameter, the parameter being associated with processing of asubstrate. This may mean that the parameter is affected or controlledduring processing of the substrate, or that the parameter results fromthe processing. The processing is carried out by the devices and mayinclude various different processes, such as etching and annealing asdescribed below. The method may be used in relation to a fingerprint ofvarious different parameters. The parameter may be any parameteraffected or controlled during processing of a substrate. For example,the parameter may be selected from a group comprising criticaldimension, overlay, critical dimension uniformity, side-wall angle, lineedge placement, alignment, focus (which may otherwise be referred to aslevelling), pattern shift, line edge roughness, micro topology, and/oredge placement error (EPE). Additionally or alternatively, the parametermay be selected from a group comprising a shape description of a featuresuch as side-wall angle, resist height, and/or contact hole ellipticity.Additionally or alternatively, the parameter may be selected from agroup comprising a processing parameter such as a coating thickness,optionally bottom anti-reflective coating thickness and/or resistthickness. Additionally or alternatively, the parameter may be selectedfrom a group comprising a processing parameter such as an opticalproperty of a coating, which may optionally indicate a measure ofabsorption, such as refractive index and or extinction coefficient.Additionally or alternatively, the parameter may be selected from agroup comprising a parameter determined from substrate measurements,such as yield parameters, optionally defects and/or electricalperformance. The method may be applied to any one of these parametersand could be used on multiple parameters depending on which of theparameters are of most interest or importance to a particular user.

In this embodiment there may be at least two different classes ofdevices. Only two classes are described, but additional classes may alsobe provided. The devices in a particular class may be used to carry outsimilar functions, such as etching or annealing as in the exampleillustrated in FIG. 2. In other words the devices in a given class areof the same type. For example, as shown in FIG. 2, at least two of theclasses have at least two devices. This means that there may be at leasttwo devices of the same type in at least two of the classes. Thesubstrate may only be processed by one device in a first class ofdevices and/or one device in a second class of devices, e.g. thesubstrate might be processed by only one of the devices in at least oneclass, or even only one of the devices in each class.

In the example depicted in FIG. 2, there are several different classesof devices shown. For example, the etching devices 122A, 122B and 122Care each in a first class of device and the annealing devices 124A, 124Band 124C are each in a second class of device. As shown in FIG. 2, thereare three etching devices and three annealing devices. Different numbersof classes, different numbers of one or more devices in each class anddifferent combinations of class and numbers in each class may beprovided.

As there is a plurality of devices, it is desirable to determine theeffect of an individual device. Thus, in an embodiment, the method isfor determining a contribution of a single device out of the pluralityof the devices. In this way, the impact of a single device can becalculated. As described, the parameter may relate to many things butmost generally, the parameter is associated with processing of asubstrate. Thus, the parameter is likely affected by processing thesubstrate and is thus likely to be affected in different ways by thedifferent devices in the plurality of devices.

The method further comprises obtaining parameter data and usage data.The parameter data is based on measurements for multiple substrateshaving been processed by the plurality of devices. In further detail,the parameter data relates to measurements corresponding to theparameter for substrates processed by the plurality of devices. Theusage data indicates which of the devices out of the plurality ofdevices were used in processing of a substrate. In other words, theusage data indicates which devices have been specifically used forprocessing each substrate. The usage data thus provides an indication ofwhich individual devices, e.g. which particular device within a class ofdevices, has been used for processing a substrate. The devices used forprocessing the substrate can include the devices described above andshown in FIGS. 1 and 2, and include one or more devices within thelithographic apparatus 100 used for processing the substrate, e.g. forexample the substrate table used for exposing (in a dual stagelithographic apparatus).

The method further comprises determining the contribution using theusage data and parameter data. Thus, the method determines thecontribution from one of the devices out of the plurality of devices toa fingerprint of a parameter associated with processing the substrate byusing the above described data. This can be done in a variety ofdifferent ways.

In an embodiment, the method may further comprise using a matrix todetermine the contribution. The matrix may refer to an array ofquantities or expressions in rows or columns. The matrix may be part ofa matrix equation which may be solved to determine unknown values in theequation. The method may comprise a step of determining a matrix usingthe usage data. The step of determining the contribution may comprisesolving an equation comprising the matrix and using the parameter data.The matrix may represent the devices used to process multiplesubstrates. Thus the matrix may be used to define a relationship betweenthe contribution of a device to a fingerprint of a parameter and theparameter data. This embodiment may be used with any matrix solutionmethod or variation thereof. As will be described in the examples below,the matrix equation may be solved to determine the contribution of adevice out of a plurality of devices by multiplying a transformedversion of the matrix with the parameter data, optionally wherein thetransformed version of the matrix is the inverse of the matrix.

Determining the matrix may be done in a variety of different ways.Generally, in an embodiment, each row of the matrix represents asubstrate having been processed by at least one device in a first classof devices and at least one device in a second class of devices. Thus,each row represents a substrate having been at least partiallyprocessed. This means that each substrate which provides measurementsbeing used to determine the impact of the devices has a singlecorresponding row in the matrix. This may correspond to the substratehaving passed through at least one of the first class of devices and atleast one of the second class of devices. In the example of FIG. 2, thefirst class of devices are the etching devices 122A, 122B and the secondclass of devices are the annealing devices 124A, 124B. Each column inthe matrix may represent one device of the plurality of devices. Thesubstrate may only be processed by one device in any particular class,e.g. one device in the first class of devices and/or one device in thesecond class of devices.

As the substrate is processed, it will pass through different types ofdevice, i.e. devices in different classes. Thus, the substrate may passthrough at least one device in a variety of classes. The substrate maypass through at least one device in each of the classes. For eachsubstrate, data can be collected and/or obtained to indicate whichdevices have been used to process that substrate. As described, this isthe usage data. The usage data can be used to generate the matrix. Therow of the matrix has a non-zero entry corresponding to each of thedevices used to process that particular substrate. In other words, foreach substrate, there is an entry in the matrix indicating whichparticular device from the plurality of devices has been used inprocessing that substrate. The entry may be a value, such as 1, or maybe a sub-design matrix (which will be described in further detailbelow). In an example, the model may be solved for individualmeasurement locations, wherein the model per location is:

x _(i) =b _(etchA,i) +b _(annealB,i)  (1)

where i indicates the measurement number, i.e. indicates which substratethe measurement relates to. In this example, the parameter data x isbased on at least one measurement from a substrate that has beenprocessed by an etching device and an annealing device. A and B canindicate any of a particular etching and annealing device respectively.For example, A may be replaced with 1, and etch1 may correspond to thefirst etching device 122A in FIG. 2, and B may be replaced with 2, andanneal2 may correspond to the second annealing device 124B in FIG. 2.The symbols are not in bold because the parameters (e.g. the b's) aresingle values, and can be interpreted as decomposed measurements. Alldecomposed measurements may be fitted with a fingerprint model, ifdesired.

For individual measurement locations for multiple substrates, themultiple substrates may each be processed by at least one device. Tosolve equation (1), multiple equations are provided for differentsubstrates, i.e. having different values of i. A matrix can beformulated based on the model for the multiple substrates and equation(1) can apply to each substrate. The parameter data based onmeasurements for each substrate at a particular location can be writtenas a vector, wherein x=[x₁, x₂, x₃ . . . x_(n)]. Equations which areequivalent to equation (1) for each substrate can be combined as amatrix formulation. The matrix formulation of a linear model forestimating the fingerprint of a parameter can be represented as:

x=M·b  (2)

wherein the symbols are in bold because they refer to vectors and amatrix. In this example, x is a vector based on measurements formultiple substrates having been processed by the plurality of devices,i.e. x is the parameter data, b is a vector of parameters of the linearmodel, and M is a matrix. A substrate is represented by a row of matrix,M, so the number of rows of the matrix, M, is the same as the size ofthe vector x. Each column may contain a basis function for one parameterin b, evaluated for all measurements, and each row may contains allbasis functions evaluated based on measurements for one substrate.

In an example, six substrates (e.g. x₁, x₂, x₃, x₄, x₅, x₆) areprocessed by a first etching device 122A, a second etching device 122Bor a third etching device 122C, and a first annealing device 124A, asecond annealing device 124B or a third annealing device 124C as shownin FIG. 2. The first substrate is processed using the first etchingdevice 122A and the first annealing device 124A. The second substrate isprocessed using the second etching device 122A and the second annealingdevice 124B. The third substrate is processed using the first etchingdevice 122A and the third annealing device 124C. The fourth substrate isprocessed using the third etching device 122C and the first annealingdevice 124A. The fifth substrate is processed using the second etchingdevice 122B and the third annealing device 124C. The sixth substrate isprocessed using the third etching device 122C and the second annealingdevice 124B.

Each substrate has an equation corresponding to equation (1), and thecombinations from the devices used for the different substrates can beapplied in a matrix formulation as in equation (2). Thus, the matrixformulation of equation (2) may be written out in full for this exampleas follows:

$\begin{matrix}{\begin{pmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4} \\x_{5} \\x_{6}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 0 & 1 & 0 & 0 \\0 & 1 & 0 & 0 & 1 & 0 \\1 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 1 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 1 & 0\end{pmatrix} \cdot \begin{pmatrix}b_{{etch}1} \\b_{{etch}2} \\b_{{etch}3} \\b_{anneal1} \\b_{anneal2} \\b_{anneal3}\end{pmatrix}}} & (3)\end{matrix}$

As described, each row of matrix, M, indicates a substrate and eachcolumn represents one of the devices. Thus, in this example, the matrix,M, may be generated to represent the first substrate in a first row, thesecond substrate in a second row and the third substrate in the thirdrow, and so on. Furthermore, the first column may correlate to the firstetching device 122A, the second column may relate to the second etchingdevice 122B, the third column may relate to the third etching device122C, the fourth column may relate to the first annealing device 124A,the fifth column may relate to the second annealing device 124B and thesixth column may relate to the third annealing device 124C.

The first substrate should have non-zero entries corresponding to thefirst etching device 122A and the first annealing device 124A. Thus, thefirst substrate (in the first row) has a non-zero entry in the firstcolumn and fourth columns of the matrix, M. As the first substrate wasnot processed by the second etching device 122B, third etching device122C, the second annealing device 124B or the third annealing device124C, the first row has zero entries in the second, third, fifth andsixth columns.

The second substrate should have non-zero entries corresponding to thesecond etching device 122B and the second annealing device 124B. Thus,the second substrate (in the second row) has a non-zero entry in thesecond and fifth columns of the matrix, M. As the second substrate wasnot processed by the first etching device 122A, third etching device12C, the first annealing device 124A or the third annealing device 124C,the second row has zero entries in the first, third, fourth and sixthcolumns.

The third substrate should have a non-zero entry corresponding to thefirst etching device 122A and the third annealing device 124C. Thus, thethird substrate (in the third row) has a non-zero entry in the first andsixth columns of the matrix, M. As the third substrate was not processedby the second etching device 122B, the third etching device 122C, thefirst annealing device 124A or the second annealing device 124B, thefirst row has zero entries in the second, third, fourth and fifthcolumns.

The entries in the matrix for the fourth, fifth and sixth substrates aredetermined in the same way. In this way, the usage data can be used togenerate a matrix, M. The matrix, M, clearly indicates which of theplurality of devices have been used for processing a substrate. Thus amatrix, M, is determined as shown below which corresponds to the matrixin equation (3):

$\begin{matrix}{M = \begin{pmatrix}1 & 0 & 0 & 1 & 0 & 0 \\0 & 1 & 0 & 0 & 1 & 0 \\1 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 1 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 1 & 0\end{pmatrix}} & (4)\end{matrix}$

The same equations as above may be applied for more complicated models,such as when the parameter data is based on measurements which arepreprocessed. In an example, the model of contributions from differentdevices can be written as follows:

x=M _(A) ·b _(A) +M _(B) ·b _(B)  (5)

wherein A and B are labels of a first and second device. The model canbe different for the devices A and B, and can have different numbers ofparameters, which would lead to different sizes of the vectors b, anddifferent number of columns of the matrices, M. As before, x is also avector. This example corresponds to the equation (1) but usingsub-design matrices.

Solving equation (5) requires more than one set of measurements. Thus,the parameter data is based on measurement for multiple substrates,which as shown in equation (6) may be a first and second substrate:

$\begin{matrix}\left\{ \begin{matrix}{x_{1} = {{M_{A,1} \cdot b_{A}} + {M_{B,1} \cdot b_{B}}}} \\{x_{2} = {{M_{A,2} \cdot b_{A}} + {M_{B,2} \cdot b_{B}}}}\end{matrix} \right. & (6)\end{matrix}$

wherein the numerical subscripts indicate the substrate which has beenprocessed, and A and B are labels of the devices used to process thatsubstrate, as in equation (5). Each equation in (6) may relate to amodel/fingerprint for one substrate, i.e. relating to multiple locationson a single substrate.

Equation (6) can be rewritten as:

$\begin{matrix}{\begin{pmatrix}x_{1} \\x_{2}\end{pmatrix} = {\begin{pmatrix}M_{A,1} & M_{B.1} \\M_{A,2} & M_{B.2}\end{pmatrix}.\begin{pmatrix}b_{A} \\b_{B}\end{pmatrix}}} & (7)\end{matrix}$

which shows how the design matrix may be composed of multiple sub-designmatrices, e.g. M_(A,1), M_(A,2) . . . etc. Each sub-design matrix may begenerally written as M_(X,i), where i indicates the substrate and Xindicates the device use on the substrate. The sub-design matrix,M_(X,i) is a matrix within the design matrix, M. This mechanism ofcombining two linear models can be extended to include more contributormodels for additional substrates and/or devices. The sub-design matrixmay be a zero entry sub-design matrix or a non-zero entry sub-designmatrix. A zero entry sub-design matrix may correspond to the devices outof the plurality of devices not used to process the substrate. The zeroentry sub-design matrix may be a matrix comprising only 0 entries. Thesub-design matrices in a single matrix may all be the same size, suchthat the zero-entry sub-design matrix and the non zero-entry sub-designmatrix are the same size. A non-zero entry sub-design matrix maycorrespond to each of the devices used to process the substrate. Thenon-zero entry sub-design matrix may be based on a modelled contributionto the fingerprint from the respective device and substrate. In otherwords, the non zero-entry sub-design matrix may include informationmodelling the contribution on a particular substrate from a particulardevice. The zero entry sub-design matrices may be used in place of thezero entries in the matrix described above and the non zero-entrysub-design matrices may be used in place of the non zero entries in thematrix described above.

In the present example, for every substrate, a model with one term foreach class of devices used on that substrate is created, for example:

$\begin{matrix}{\begin{pmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4}\end{pmatrix} = {\begin{pmatrix}M_{{{etch}1},1} & 0 & M_{{{anneal}1}\text{.1}} & 0 \\0 & M_{{{etch}2},2} & 0 & M_{{{anneal}2}\text{.2}} \\M_{{{etch}1},3} & 0 & 0 & M_{anneal{2.3}} \\0 & M_{{{etch}2},4} & M_{{{anneal}1}\text{.4}} & 0\end{pmatrix}.\begin{pmatrix}b_{{etch}1} \\b_{{etch}2} \\b_{anneal1} \\b_{anneal2}\end{pmatrix}}} & (8)\end{matrix}$

In this example, etch indicates a first class of devices, i.e. etchingdevices, and anneal indicates a second class of devices, i.e. annealingdevices. The 0 matrices have dimensions which correspond to thematrices, M_(X,i).

Each of the plurality of devices may be used to process at least one ofthe multiple substrates. In other words, to solve the equationcomprising the matrix to determine the contribution from a device out ofa plurality of the devices, the device must have been used on at leastone of the substrates in the matrix. The fewer combinations of devicesthere are, the greater the uncertainty in the resulting determination.

Although the etching and annealing devices are used as examplesdescribed above, any different type of device (as well as any numbers ofdevices in one type and any number of different types of device) may beused. The device may be any device which impacts a parameter describedabove. Thus the different types of device could include etching devices,deposition tools, substrate tables, polishing devices (such aschemical-mechanical planarization devices), annealing devices (such asrapid thermal annealing devices), cleaning devices, coating devices(such as those used to apply a resist), developing devices, trackdevices, implantation devices and/or baking devices. As would beunderstood, any combination of devices may be applicable and may be usedon the substrate. Determining the contribution may increase incomplexity as the number of devices increases but would still bepossible using the matrix method described.

The parameter data may relate to the measurements in different ways (themeasurements may otherwise be referred to as measurement data). In anexample, the parameter may be the same as the measurements. In otherwords, no pre-processing of the measurements may be carried out and thecontribution may be determined using the parameter data without initialprocessing. Alternatively the measurements could be processed using avariety of different methods to provide the parameter data. In otherwords, the parameter data is based on processed measurements. Forexample, the measurements can be processed using principal componentanalysis or fitting a model to the measurements (such as a polynomialmodel or a linear model, e.g. using Zernike analysis). Using Zernikeanalysis may use sub-design matrices as described above. Principalcomponent analysis (e.g. using statistical or spatial correlation) mayhave an advantage of filtering noise. Statistical correlation may beused which may only be based on major fingerprints which suppressesdifferences that are small or not correlating between substrates. Theparameter data which is based on processed measurements may result insub-design matrices as described in relation to equations (5) to (8).

Different types of fit may be used and they may be effective forsuppressing noise. Thus, the parameter data may be a model based onmeasurement for the substrates having been processed by the plurality ofdevices. This may decrease the number of measurements needed while stillproviding usable parameter data with accuracy to a desired degree.

The equation comprising the matrix may be solved using a variety ofdifferent methods. Most generally, the matrix equation can be solved bymultiplying a transformed version of the matrix with the parameter data.For example, equation (2) above can be solved using standard linearalgebra techniques, e.g. the equation comprising the matrix can besolved using a least squares fit:

b=(M′·M)⁻¹ ·M′·x  (9)

and a similar method may be applied to equation (7). An advantage ofusing a least squares fit is that this may be quicker than other methodsand it can be solved without explicit matrix inversion, using e.g.Penrose pseudo-inverse, or QR decomposition. Additionally, the aboveequations, e.g. equations (2) and/or (7), may be adapted to include anindication of noise variance such that noise variance has a reducedeffect. However, it won't work on an undetermined system and there maybe overfitting. In case the M′M matrix is almost singular, alternativetechniques exist to make the problem more easily solvable, e.g. singularvalue decomposition (SVD).

Singular value decomposition may be used based on the following theoremin which:

M=U _(m×m) S _(m×k) V _(k×n)  (10)

where U and V are each unitary matrices, U^(T)U=I_(n×n) andV^(T)V=I_(p×p) (i.e. U and V are orthogonal), and S is a diagonalmatrix. The subscript m and n values represent the number of rows andcolumns of each matrix respectively, i.e. U is an m by m matrix and S isan m by k matrix. This can be applied to the above equations such that:

x _(m×p) =U _(m×m) S _(m×k) V _(k×n) b _(n×p)

This equation can be solved to provide the following solution:

b=V _(k)(S _(k) ^(T) S _(k))⁻¹ S _(k) ^(T) U ^(T) x  (12)

Part of the problem is that the matrix is likely to be overdetermined,meaning that there is not necessarily one simple solution. This meansthat inverting the matrix often won't work. For any solution, aninfinite number of alternative solutions may be created by adding anarbitrary number or vector to all b or b of one group, whilesimultaneously subtracting the same number of vector from the b or b ofanother group.

One useful way to overcome this is to make the average contribution of aclass of devices zero, and the average fingerprint is in a new b or b.

Referring to the above equations, this means that an additionalb_(global) is added to the model and there will be the following twoadditional constraints which can be handled during solving the model:

b _(etch1) +b _(etch2)=0  (13)

b _(anneal1) +b _(anneal2)=0  (14)

This example of including additional constraints is based on adaptingequation (8) described above.

Alternatively, the additional constraints may be made explicit in thematrix, by removing one of the columns of each group, and adding −1 inthe remaining columns of that group, in case the measurement orsubstrate has the removed columns contributor label. In addition, theremoved contributors' b or b are also removed. In the next example,based on equation (3), the last contributor of each group is removed:

$\begin{matrix}{\begin{pmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4} \\x_{5} \\x_{6}\end{pmatrix} = {\begin{pmatrix}1 & 1 & 0 & 1 & 0 \\1 & 0 & 1 & 0 & 1 \\1 & 1 & 0 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1 & 0 \\1 & 0 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 0 & 1\end{pmatrix} \cdot \begin{pmatrix}b_{global} \\b_{{etch}1} \\b_{{etch}2} \\b_{anneal1} \\b_{anneal2}\end{pmatrix}}} & (15)\end{matrix}$

After the system is solved, b_(etch3) may be calculated as−(b_(etch1)+b_(etch2)); likewise, b_(anneal3) is−(b_(anneal1)+b_(anneal2)).

The equation comprising the matrix can also be solved using a leastsquares fit based on regularization. Optionally, an L-curve methodand/or leave-one-out cross validation method may be used. Regularizationtechniques may be used to prevent overfitting. These methods may bebeneficial in suppressing randomness, which might include measurementnoise and/or other contributors. Use of an L-curve method is describedin “The L-curve and its use in numerical treatment of inverse problems”by P. C. Hansen in Computational Inverse Problems in Electrocardiology,ed. P. Johnston, Advances in Computational Bioengineering(https://www.sintef.no/globalassets/project/evitameeting/2005/Icurve.pdf),which is hereby incorporated by reference in its entirety.Regularization generally limits the range of fitted parameters, e.g.reducing high frequencies, and it still allows use of the full model,e.g. the mixing model, but implicitly suppresses noise in themeasurements. For example, regularization techniques, such as ridgeregression, can be used to reduce or minimize fit error. Such techniquesmay be used based on the following equation:

b=(M ^(T) M+λI)⁻¹ M ^(T) x  (16)

In brief, for a set of lambdas (λ), the procedure comprises a step ofleaving out one substrate for all combinations (so substrates 1-19,substrates 1-18+20, etc.) and solving the model in equation (16) with aparticular lambda. Further steps include calculating the fit error onthe left-out substrate, calculating the average fit error, plotting theaverage fit error against lambda and choosing a lambda with the smallestfit error. The equation comprising the matrix can also be solved usingBayesian statistics which has advantages of avoiding overfitting, priorknowledge can be systematically incorporated and a distribution of theresult can be provided rather than a single point. However, heavycomputational processing may be required and there may be large memoryrequirements.

In the case of multidimensional measurements (e.g. overlay), the abovemethods can be applied separately on x and y.

In the above example, an entry in the matrix may simply be 1 (as thenon-zero entry) or a zero indicating the specific device used to processa particular substrate. However, the matrix may comprise at least onesub-design matrix as shown in equations (5)-(8). In other words, anentry in the matrix may comprise a further matrix. For example, thesimpler matrix may be used when the parameter is determined at anindividual location and the matrix may be more complex, for example byusing sub-design matrices when processing multiple locations. Thecontent of the sub-design matrix content can vary, for example differentsub-design matrices may be used for different classes, e.g. anoffset/curvature for one class of devices and global for another classof devices.

In the above described one or more embodiments, a matrix can bedetermined and then used to determine the contribution of a device, outof a plurality of devices, to a fingerprint of a parameter. Asdescribed, there are several ways in which the matrix can be determined.The above-described method seeks to find incorrect disturbance sourceswhich negatively impact on parameters of printed substrates W.Measurements of the substrate W can be decomposed based on context datato work out the contribution of a specific device. In this way, themethod can be used to determine what affects a parameter of a printedsubstrate with relatively good accuracy. In this way, negativecontributions to a parameter from a specific device can be accountedfor, i.e. minimized and/or corrected.

A sufficiently information-rich data set may be used to properlydecompose the fingerprint of a parameter to determine the contributionof a particular device. This may mean that a relatively large set ofdata is needed, which can be costly. In other words, a possible issue isthat a large number of substrates W may need to be processed in avariety of different ways and then measured in order to determine thespecific fingerprint for any one device. Additionally, there may be arelatively large amount of redundant data in the data set. In order toget the desired data set, sampling periodically/randomly, etc. could beused to reduce the number of substrates which need to be processed andmeasured. However, a better data set could be provided by using a morefocused method as described below. The method might indicate wheremeasurements should be carried out and on how many substrates W. Thus,the method may comprise steps to improve or optimize the determinationof the matrix so that fewer substrates W are used to determine thecontribution of the different devices.

In further detail, the matrix can be determined in a specific way toprovide advice about which substrates W should be measured and/or whichdevices should be used to process substrates to measure a useful numberof substrates in the shortest amount of time and/or to advise how toextend a data set with a reduced or minimal number of substrates, as anaddendum to a data set already gathered, e.g. by determining whichdevices to use to process selected additional substrates.

Generally, the method for determining the matrix is for selecting athread of process steps applied to multiple substrates. This means thatthe matrix may be used to select which combination of devices should beused to process individual substrates in a selected group of substrates.The method comprises selecting a first thread (e.g. a first combinationof devices for processing a substrate) which may correspond to a row inthe matrix. The method may further comprise selecting at least onefurther thread (e.g. at least one further combination of devices forprocessing at least one further substrate) based on an expectedimprovement with which a characteristic of a process step associatedwith the first thread and the second thread can be determined.

In further detail, the method for determining the matrix as describedherein may further comprise the following steps, which are desirablyprovided in the following order:

a. generating a first matrix, N, representing the possible combinationsof the plurality of devices used to process a substrate W;

b. determining a vector, n_(i), for each row, i, of the first matrix, N;

c. calculating a delta covariance matrix, ΔY_(i) for each row, whereinthe delta covariance matrix is n_(i) ^(T)n_(i);

d. selecting a row, i, from the first matrix, N, and storing theselected row as a second matrix, M;

e. determining if a stopping criteria has been met, wherein:

if the stopping criteria is not met, continuing with step f using anupdated first matrix, N, in which the selected row is removed; and

if the stopping criteria is met, using the second matrix, M, as thedetermined matrix;

f. calculating the pseudo-determinant corresponding to each remainingrow of the updated first matrix, N;

g. determining a row with a preferred pseudo-determinant, updating thesecond matrix, M, to include the row with the preferredpseudo-determinant, and updating the first matrix, N, by removing therow with the preferred pseudo-determinant;

h. determining if a stopping criteria has been met, wherein:

if the stopping criteria is not met, returning to step f using anupdated first matrix, N, in which the row with the preferredpseudo-determinant is removed; and

if the stopping criteria is met, the updated second matrix, M, is usedas the determined matrix.

Determining the matrix in this way means that it might be possible todetermine how the parameters of a printed substrate W have been affectedwhile using a low or minimal number of test substrates. This methodprovides a sampling scheme optimization which allows selection ofmetrology targets to generate informative measurements with low orminimal metrology load. Therefore, using this method means that themetrology load (i.e. the number of substrates to be measured) can bereduced while still providing a matrix which is beneficial to the user.

The above described method provides guidance about which substrates,having been processed by specific devices, to measure based on thecontents of the second matrix, M. This is beneficial in that it reducesor minimizes measurement effort while enabling good accuracy in theestimated contribution. This provides a cost-effective alternative tomeasuring additional substrates.

In the present method, substrates W can be processed according to anoptimal, possibly limited, combination of threads (i.e. a specificcombination of devices for processing individual substrates) which meansthat only measurements from these substrates can be used. This selectionof the most informative threads yields accurate context specificfingerprint determination. The covariance analysis techniques used inthe above described steps do not generally require a large amount ofcomputational processing to execute. Furthermore, this method isadvantageous in that if a large number of substrates W have beenprocessed for a large number of threads, the method may be used to guidewhich performance parameter measurements should be carried out todetermine context specific fingerprint information.

It should be noted that the first matrix, N, uses usage data indicatingwhich of the devices out of the plurality of the devices were used inthe processing of each substrate. Thus, the first matrix, N, mayrepresent all the possible combinations of the plurality of devices usedto process a substrate.

The first matrix N may otherwise be referred to as a context mixingmatrix, i.e. a matrix holding all possible context paths a substrate cantraverse, i.e. all the different possible combinations of devices whichmay be used to process a substrate. The vector n_(i) for the ith rowvector may otherwise be referred to as the ith row vector. When thefirst row is added to the second matrix, M, this may be referred to asinitializing a sampling scheme, wherein a sample of the first matrix, N,is used to form the second matrix, M.

The size of the delta covariance matrix, ΔY_(i) for each row may bebased on the size of the first matrix, N. If the matrix N is ofdimension n_a by n_b, then ΔY_(i) is of size n_b by n_b, i.e. a squarematrix with the same number of rows and columns. As described, n_b maybe the amount of devices, whereas n_a may indicate the plurality ofcombinations of devices.

When the rows are selected from the first matrix, N, and stored in thesecond matrix, M, the new rows may be appended to the second matrix, M,at the bottom of the second matrix, M. Thus, in this way, the rows canbe transferred from the first matrix, N, to the second matrix, M, andthe rows being added to the second matrix, M, can be added in an orderedway. This is advantageous because if the second matrix, M, needs to betruncated at a later date, for example if a user needs to further limitthe number of substrates tested, then the second matrix, M, can bereduced while maintaining optimum rows at the top of the second matrix,M. Additionally, if the rows are added to the second matrix, M, in anordered way, then this may allow a user to more easily evaluate thepreferred size of the experiment. For example, a certain parameter ofinterest may be compared to the number of substrates/differentcombination of processing devices used on the substrates. For example,the parameter may be plotted on a function of the “size” of theexperiment. This could be used to check how many rows of M are needed topass a certain threshold, i.e. meet a stopping criteria. Then thisdenotes the “size” of the experiment (e.g. the number of substrates tobe sampled).

The method described above essentially uses steps f to h to probe eachof the rows remaining in the first matrix, N, to determine which of therows is the most valuable. The most valuable row is then added to afurther matrix. The further matrix is the second matrix, M. In otherwords, the second matrix, M, can be used in the linear equation in theone or more embodiments described above. The second matrix, M, can belater used to determine the contribution of a device out of a pluralityof devices. When a row is removed from the first matrix, N, the processis repeated and the remaining rows are then probed to determine which ofthe remaining rows is the most valuable. The next most valuable row isthen added to the second matrix, M. In this way, the most valuable rowsof the first matrix, N, can be added to the second matrix, M, until astopping criteria is met. The stopping criteria can be selected orpredetermined depending on a variety of different options as describedbelow.

As described above, the pseudo-determinant for each row is calculated todetermine the row with the preferred pseudo-determinant. Thepseudo-determinant is calculated by adding the delta covariance matrix,ΔYi, for each row to a third matrix, Y, wherein the third matrix, Y, isthe covariance of the second matrix, M. Further to this, the determinantof the sum of the third matrix, Y, and the delta covariance matrix forthe row, ΔYi can then be calculated. Thus, the determinant of Y+ΔYi iscalculated. In every iteration, i.e. for each row i out of the firstmatrix, N, the pseudo determinant of Y+ΔYi is evaluated. This equates tothe sum of the covariance of the current second matrix, M, and ΔYi (forrow i of the first matrix, N). This calculation is carried out for allrows i of the first matrix, N. The third matrix, Y, is kept track of,and represents the covariance of the second matrix, M. Thus, wheneverthe second matrix, M, is updated, the third matrix, Y, is also updatedaccordingly.

A row of the first matrix, N, may represent a possible combination ofthe plurality of devices used to process a substrate. A column of thefirst matrix, N, may represent one of the plurality of devices. In thiscase, a row of the first matrix, N, may have a non-zero entrycorresponding to each of the plurality of devices used to process thesubstrate represented by that respective row. Similarly, the firstmatrix, N, may have a zero entry corresponding to each of the pluralityof devices not used to process the substrate represented by thatrespective row.

It is described above that the preferred pseudo-determinant isdetermined and then included in this second matrix, M. The preferredpseudo-determinant may otherwise be referred to as the optimumpseudo-determinant. The preferred pseudo determinant may be consideredto be the pseudo-determinant with the highest value. In other words, therow n out of the first matrix, N, which attained the highest pseudodeterminant may added to the second matrix, M. One or more other meansmay be used. For example, the trace of inv(Y+ΔY_(j)) could be used (thetrace is the sum over the diagonal elements). In this case, the row withthe minimum value of the trace may have the preferredpseudo-determinant. Alternatively, a value of v_j could be minimizedwith v_j=sqrt(x_j*inv(Y+ΔY_(j))*x_j′), with x_j being the j_th row outof the combined [M;N] matrix. In this case, the row with the maximumvalue of v_j may have the preferred pseudo-determinant.

After the delta covariance matrix for each row is calculated, a row isselected (as the first selected row) from the first matrix, N. The rowthat is first selected may be the first row of the first matrix, N. Thisis not necessary, however, it simplifies the process in that the firstselected row is simply the first row of the first matrix, N.

As described above, the process for determining the matrix may bestopped if a stopping criteria is met. The stopping criteria could bebased on one or more of a variety of different selected criteria. Thestopping criteria may otherwise be referred to as a key performanceindicator. The stopping criteria may have a predetermined or selectedvalue. Although, this value may be altered, for example, depending onthe selection of a user.

For example, the stopping criteria may be met if the number of rows usedin the second matrix, M, reaches a predetermined value. Additionally oralternatively, the stopping criteria may be met if all rows of the firstmatrix, N, are used in the second matrix, M. Additionally oralternatively, the stopping criteria may be met if a value of aperformance parameter reaches a predetermined value. The performanceparameter may be any parameter relating to the substrate. For example,the performance parameter may be one or more selected from: i. criticaldimension, overlay, critical dimension uniformity, line edge placement,alignment, focus, pattern shift, line edge roughness, micro topology,and/or edge placement error; and/or ii. a shape description of afeature, such as side-wall angle, resist height, and/or contact holeellipticity; and/or iii. a processing parameter such as a coatingthickness such as bottom anti-reflective coating thickness and/or resistthickness, and/or an optical property of a coating which may indicate ameasure of absorption, such as refractive index and/or extinctioncoefficient; and/or iv. a parameter determined from substratemeasurements, such as a yield parameter, such as defects and/orelectrical performance. It will be understood that a user may selectother stopping criteria which may be used in addition or as analternative to the stopping criteria described here.

When all the rows of the first matrix N are used, the rows may still beeffectively sorted in the second matrix, M, by relevance because themost or more optimal rows may be provided towards the top of the secondmatrix, M. Thus, the list of rows in the second matrix, M, could belater truncated to a reduced size. In other words, when all of the rowsof the first matrix, N are used in the second matrix, M, the resultingmatrix could be later truncated resulting in the same effect as otherstopping criterion, such as a predetermined data set.

The above steps of the method include various determining, calculatingand selecting steps. It will be understood that the data used during andafter each of these steps may be stored in an appropriate data storagemedium (e.g., semiconductor memory, magnetic or optical disk).

In a further embodiment, the contribution is determined withoutrequiring the use of a matrix (although a matrix equation may optionallybe used). This embodiment is essentially the same as the earlierembodiments except for the use of the matrix. This embodiment maycomprise the step of analyzing variation of the parameter data using theusage data, i.e. carrying out “Analysis of Variation” (ANOVA) on themeasurement data. In this embodiment, the general idea is to evaluatethe variation between the measurement data when grouping the measurementdata according to the usage data. Thus, this method comprisesdetermining the contribution to the parameter for each device bygrouping the measurement data using the analyzed variation. In additionthe contribution to a fingerprint from the various parameters can beassessed using a fingerprint model. This is done most efficiently with amodel of the fingerprint based upon principal component analysiscoefficients (also called loading) of the data.

For determining a contribution of a device to a fingerprint of aparameter, the method may comprise using the ANOVA procedure (e.g.Matlab “anovan”) per measurement location, and retrieving for each classof device the difference between the mean of the group measurements perdevice and the global mean, and taking that as the decomposedmeasurements. All decomposed measurements may then be fitted with afingerprint model, if desired.

The methods of the embodiments herein may be used to control a device.At least one of the devices may be controlled based on the determinedcontribution to the fingerprint of the parameter of that device. Thus,the determined contribution for a device may be used to alter the way inwhich the processing of a substrate is carried out by that device. Inother words, any error or variation in fingerprint which is undesirablyinduced by a specific device may be reduced or eliminated by controllingthe device based on its determined parameter contribution. Additionallyor alternatively, other devices including a litho tool 100 may becontrolled based on the determined parameter contribution. As described,the parameter may be many different things. The parameter may be anyparameter that directly or indirectly influences the control applied bya device.

It is described above that a row of the matrix may represent asubstrate, and in particular, a substrate having been processed by atleast one device in a first class of devices and at least one device ina second class of devices, and a column represents one of the pluralityof devices. However, it will be understood that the rows and columns maybe switched. Thus, a row of the matrix may represent one of theplurality of devices and a column may represent a substrate. In otherwords, the matrix described above may be transposed.

Generally, the methods described above are used for at least twoclasses, and at least two devices in at least two classes because thismakes the matrix overdetermined. Theoretically, at least one class mayhave only one device, and this would still be useful for determining theeffect of the plurality of devices. At least one class may comprise morethan two different devices, as shown in the examples described above inrelation to equations (7) and (8). Theoretically, there may only be oneclass of devices and multiple devices within that class. In this case,it would still be useful, e.g. to obtain an average fingerprint based onthese devices. Alternatively, although the description refers to a firstclass and a second class, there may be more than two different classes.The classes may include the types of classes described above, acombination of at least one of the described classes with at least oneadditional class, or different classes may be provided. There may bedifferent numbers of classes and different numbers of devices in eachclass.

In an embodiment, a system is provided comprising a processor configuredto determine a contribution from a device out of a plurality of devicesto a fingerprint of a parameter, the parameter being associated withprocessing of a substrate. The processor is configured to carry out themethod according to any of the embodiments herein. The processor may bepart of, or connected to, either the automated process control (APC)system and/or the supervisory control system.

The processor may be configured to determine a contribution from adevice out of a plurality of devices to a fingerprint of a parameter,the parameter being associated with processing of a substrate, theprocessor being configured to: obtain parameter data and usage data,wherein the parameter data is based on measurements for multiplesubstrates having been processed by the plurality of devices, and theusage data indicates the devices used in the processing of eachsubstrate; and determine the contribution using the usage data and theparameter data.

In an embodiment, a computer program is provided containing one or moresequences of machine-readable instructions describing a method ofdetermining a contribution from a device out of a plurality of devicesto a fingerprint of a parameter, the parameter being associated withprocessing of a substrate. Any of the methods herein may be implementedusing a computer program containing one or more sequences ofmachine-readable instructions. There may also be provided a data storagemedium (e.g., semiconductor memory, magnetic or optical disk) havingsuch a computer program stored therein.

A program is provided for controlling determining a contribution from adevice out of a plurality of devices to a fingerprint of a parameter,the parameter being associated with processing of a substrate. Theprogram may comprise instructions for carrying out a method comprising:obtaining parameter data and usage data, wherein the parameter data isbased on measurements for multiple substrates having been processed bythe plurality of devices, and the usage data indicates which of thedevices out of the plurality of devices were used in the processing ofeach substrate; and determining the contribution using the usage dataand the parameter data. The program may comprise instructions forcarrying out the steps of any of the methods described herein.

The computer program may be executed for example within the control unitLACU of FIG. 1, or some other controller, for example within a metrologysystem that includes the metrology apparatus 140, or in an advancedprocess control system or separate advisory tool. The program mayoptionally be stored in a memory which is part of, or can be accessedby, the automated process control (APC) system and/or the supervisorycontrol system.

Further embodiments of the disclosure are disclosed in the list ofnumbered embodiments below:

1. A method of determining a contribution of a device out of a pluralityof devices to a fingerprint of a parameter, the parameter beingassociated with processing of a substrate, the method comprising:

obtaining parameter data and usage data, wherein the parameter data isbased on measurements for multiple substrates having been processed bythe plurality of devices, and the usage data indicates which of thedevices out of the plurality of the devices were used in the processingof each substrate; and

determining the contribution using the usage data and parameter data.

2. The method of embodiment 1, further comprising determining a matrixusing the usage data and wherein determining the contribution to thefingerprint comprises solving an equation comprising the matrix usingthe parameter data.3. The method of embodiment 2, wherein the matrix equation is solved todetermine the contribution of a device out of a plurality of devices bymultiplying a transformed version of the matrix with the parameter data,wherein optionally the transformed version of the matrix is the inverseof the matrix.4. The method of embodiment 2 or embodiment 3, wherein a row of thematrix represents a substrate having been processed by at least onedevice in a first class of devices and at least one device in a secondclass of devices, and the column of the matrix represents one of theplurality of devices.5. The method of any of embodiments 2 to 4, wherein a row of the matrixhas a non-zero entry corresponding to each of the devices used toprocess the substrate represented by that respective row, and a zeroentry for the devices out of the plurality of devices not used toprocess the substrate represented by that respective row.6. The method of embodiment 2 or embodiment 3, wherein a column of thematrix represents a substrate having been processed by at least onedevice in a first class of devices and at least one device in a secondclass of devices, and a row of the matrix represents one of theplurality of devices.7. The method of embodiment 2, embodiment 3 or embodiment 6, wherein acolumn of the matrix has a non-zero entry corresponding to each of thedevices used to process the substrate represented by that respective rowor column, and a zero entry for the devices out of the plurality ofdevices not used to process the substrate represented by that respectivecolumn.8. The method of any of embodiments 2 to 7, wherein determining thematrix comprises:

a. generating a first matrix, N, representing the possible combinationsof the plurality of devices used to process a substrate;

b. determining a vector, n_(i), for each row, i, of the first matrix, N;

c. calculating a delta covariance matrix, ΔY_(i) for each row, whereinthe delta covariance matrix is n_(i) ^(T)n_(i);

d. selecting a row, i, from the first matrix, N, and storing theselected row as a second matrix, M;

e. determining if a stopping criteria has been met, wherein:

-   -   if the stopping criteria is not met, continuing with step f        using an updated first matrix, N, in which the selected row is        removed; and    -   if the stopping criteria is met, using the second matrix, M, as        the determined matrix;

f. calculating the pseudo-determinant corresponding to each remainingrow of the updated first matrix, N;

g. determining a row with a preferred pseudo-determinant, updating thesecond matrix, M, to include the row with the preferredpseudo-determinant, and updating the first matrix, N, by removing therow with the preferred pseudo-determinant;

h. determining if a stopping criteria has been met, wherein:

-   -   if the stopping criteria is not met, returning to step f using        an updated first matrix, N, in which the row with the preferred        pseudo-determinant is removed; and    -   if the stopping criteria is met, the updated second matrix, M,        is used as the determined matrix.        9. The method of embodiment 8, wherein the pseudo-determinant        for each row is calculated by:

a. adding the delta covariance matrix, ΔYi, for each row to a thirdmatrix, Y, wherein the third matrix, Y, is the covariance of the secondmatrix, M;

b. calculating the determinant of the sum of the third matrix, Y, andthe delta covariance matrix for the row, ΔYi.

10. The method of embodiment 8 or embodiment 9, wherein a row of thefirst matrix, N, represents a possible combination of the plurality ofdevices used to process a substrate and a column of the first matrix, N,represents one of the plurality of devices.11. The method of embodiment 10, wherein a row of the first matrix, N,has a non-zero entry corresponding to each of the plurality of devicesused to process the substrate represented by that respective row, and azero entry corresponding to each of the plurality of devices not used toprocess the substrate represented by that respective row.12. The method of any of embodiments 8-11, wherein the optimumpseudo-determinant is the pseudo-determinant with the highest value.13. The method of any of embodiments 8-12, wherein the row firstselected in step d is a first row of the first matrix, N.14. The method of any of embodiments 8-13, wherein the stopping criteriais met if a value of a performance parameter reaches a predeterminedvalue, wherein optionally the performance parameter is one or moreselected from:

i. critical dimension, overlay, critical dimension uniformity, line edgeplacement, alignment, focus, pattern shift, line edge roughness, microtopology, and/or edge placement error; and/or

ii. a shape description of a feature, such as side-wall angle, resistheight, and/or contact hole ellipticity; and/or

iii. a processing parameter such as a coating thickness, such as bottomanti-reflective coating thickness and/or resist thickness, and/or anoptical property of a coating which may indicate a measure ofabsorption, such as refractive index and/or extinction coefficient;and/or

iv. a parameter determined from substrate measurements, such as a yieldparameter, such as defects and/or electrical performance.

15. The method of any of embodiments 8-14, wherein the stopping criteriais met if the number of rows used in the second matrix, M, reaches apredetermined value.16. The method of any of embodiments 8-15, wherein the stopping criteriais met if all rows of the first matrix, N, are used in the secondmatrix, M.17. The method of any of embodiments 2 to 16, wherein the matrixcomprises at least one sub design matrix.18. The method of embodiment 17, wherein the sub-design matrix may be azero entry sub-design matrix or a non-zero sub-design matrix, wherein azero entry sub-design matrix corresponds to the devices out of theplurality of devices not used to process the substrate and a non-zerosub-design matrix corresponds to each of the devices used to process thesubstrate, and a non-zero entry sub-design matrix is based on a modelledcontribution to the fingerprint from the respective device andsubstrate.19. The method of any of embodiments 2 to 18, wherein the equationcomprising the matrix is solved using:

-   -   i. least squares fit, optionally using regularization, and        further optionally using an L-curve method and/or leave-one-out        cross validation; and/or    -   ii. Bayesian statistics.        20. The method of any of embodiments 1 to 19, further comprising        analyzing variation of the parameter data using the usage data,        and wherein determining the contribution to the parameter for a        device comprises grouping the data using the analyzed variation.        21. The method of any of embodiments 1 to 20, wherein at least        one of the devices is controlled based on the determined        contribution to the fingerprint of the parameter of that device.        22. The method of any of embodiments 1 to 21, wherein the        parameter is one or more selected from:

i. critical dimension, overlay, critical dimension uniformity, line edgeplacement, alignment, focus, pattern shift, line edge roughness, microtopology, and/or edge placement error; and/or

ii. a shape description of a feature, such as side-wall angle, resistheight, and/or contact hole ellipticity; and/or

iii. a processing parameter such as a coating thickness, such as bottomanti-reflective coating thickness and/or resist thickness, and/or anoptical property of a coating which may indicate a measure ofabsorption, such as refractive index and/or extinction coefficient;and/or

iv. a parameter determined from substrate measurements, such as a yieldparameters, such as defects and/or electrical performance.

23. The method of any of embodiments 1 to 22, wherein the type of devicecomprises an etching device, a deposition tool, a substrate table, apolishing device, an annealing device, a cleaning device, a coatingdevice, a developing device, an implantation device and/or a bakingdevice.24. The method of any of embodiments 1 to 23, wherein the parameter datamay be the same as the measurements.25. The method of any of embodiments 1 to 23, wherein the parameter datamay be based on processed measurements, wherein the measurements areprocessed using principal component analysis or fitting of a model, suchas a polynomial model or a linear model.26. The method of any of embodiments 1 to 25, wherein each of theplurality of devices has been used to process at least one of themultiple substrates.27. The method of any of embodiments 1 to 26, wherein there are at leasttwo different classes of devices and wherein at least two classescomprise at least two devices of the same type.28. A system comprising a processor configured to determine acontribution of a device out of a plurality of devices to a fingerprintof a parameter, the parameter being associated with processing of asubstrate, the processor being configured to at least:

obtain parameter data and usage data, wherein the parameter data isbased on measurements for multiple substrates having been processed bythe plurality of devices, and the usage data indicates which of thedevices out of the plurality of devices were used in the processing ofeach substrate; and

determine the contribution using the usage data and the parameter data.

29. A program to control determining a contribution of a device out of aplurality of devices to a fingerprint of a parameter, the parameterbeing associated with processing of a substrate, the program comprisinginstructions for carrying out at least:

obtaining parameter data and usage data, wherein the parameter data isbased on measurements for multiple substrates having been processed bythe plurality of devices, and the usage data indicates which of thedevices out of the plurality of devices were used in the processing ofeach substrate; and

determining the contribution using the usage data and the parameterdata.

CONCLUSION

In conclusion, the present disclosure describes a method for determininga contribution of a device to a fingerprint of a parameter. This allowsthe contribution from a single device out of a plurality of devices tobe determined. This can be particularly useful for diagnosingperformance of the devices used for processing a substrate and/or forcontrolling the devices used for processing a substrate.

The disclosed methods allow the provision of a lithographic apparatusand methods of operating a lithographic apparatus in which acontribution of a device to a fingerprint of a parameter is determined.

Any, or all of, the steps of determining a contribution of a device to afingerprint of a parameter can be performed in any suitable processingapparatus, which may be located anywhere in the facility of FIG. 1, ormay be physically remote from the facility. Steps of the method may becarried out in separate parts of the apparatus.

The contribution, parameter data and/or usage data may be calculated inthe supervisory control system of FIG. 1, or in the litho tool controlunit LACU. They may be calculated in a remote system and communicated tothe facility afterwards. Any model and measurement data may be deliveredseparately to a processing apparatus which then combines them as part ofdetermining the contribution.

The method and variations above are described as being carried out usinga lithographic apparatus. However, one or more other apparatuses may beused. The processing step of a lithographic manufacturing process isonly one example where the principles of the present disclosure may beapplied. Other parts of the lithographic process, and other types ofmanufacturing process, may also benefit from the generation of modifiedestimates and corrections in the manner disclosed herein.

These and other modifications and variations can be envisaged by theskilled reader from a consideration of the present disclosure. Thebreadth and scope of the present invention should not be limited by anyof the above-described exemplary embodiments, but should be defined onlyin accordance with the following claims and their equivalents.

1. A method for determining a contribution of a device out of aplurality of devices to a fingerprint of a parameter, the parameterbeing associated with processing of a substrate, the method comprising:obtaining parameter data and usage data, wherein the parameter data isbased on measurements for multiple substrates having been processed bythe plurality of devices, and the usage data indicates which of thedevices out of the plurality of the devices were used in the processingof each substrate; and determining the contribution using the usage dataand parameter data.
 2. The method of claim 1, further comprisingdetermining a matrix using the usage data and wherein determining thecontribution to the fingerprint comprises solving an equation comprisingthe matrix using the parameter data.
 3. The method of claim 2, whereinthe matrix equation is solved to determine the contribution of a deviceout of a plurality of devices by multiplying a transformed version ofthe matrix with the parameter data, optionally wherein the transformedversion of the matrix is the inverse of the matrix.
 4. The method ofclaim 2, wherein a row of the matrix represents a substrate having beenprocessed by at least one device in a first class of devices and atleast one device in a second class of devices, and the column of thematrix represents one of the plurality of devices.
 5. The method ofclaim 2, wherein a row of the matrix has a non-zero entry correspondingto each of the devices used to process the substrate represented by thatrespective row, and a zero entry for the devices out of the plurality ofdevices not used to process the substrate represented by that respectiverow.
 6. The method of claim 2, wherein determining the matrix comprises:a. generating a first matrix, N, representing the possible combinationsof the plurality of devices used to process a substrate; b. determininga vector, n_(i), for each row, i, of the first matrix, N; c. calculatinga delta covariance matrix, ΔY_(i) for each row, wherein the deltacovariance matrix is n_(i) ^(T)n_(i); d. selecting a row, i, from thefirst matrix, N, and storing the selected row as a second matrix, M; e.determining if a stopping criteria has been met, wherein: if thestopping criteria is not met, continuing with step f using an updatedfirst matrix, N, in which the selected row is removed; and if thestopping criteria is met, using the second matrix, M, as the determinedmatrix; f. calculating the pseudo-determinant corresponding to eachremaining row of the updated first matrix, N; g. determining a row witha preferred pseudo-determinant, updating the second matrix, M, toinclude the row with the preferred pseudo-determinant, and updating thefirst matrix, N, by removing the row with the preferredpseudo-determinant; h. determining if a stopping criteria has been met,wherein: if the stopping criteria is not met, returning to step f usingan updated first matrix, N, in which the row with the preferredpseudo-determinant is removed; and if the stopping criteria is met, theupdated second matrix, M, is used as the determined matrix.
 7. Themethod of claim 2, wherein the matrix comprises at least one sub designmatrix.
 8. The method of claim 7, wherein the sub-design matrix may be azero entry sub-design matrix or a non-zero entry sub-design matrix,wherein a zero entry sub-design matrix corresponds to the devices out ofthe plurality of devices not used to process the substrate and anon-zero sub-design matrix corresponds to each of the devices used toprocess the substrate, wherein a non-zero entry sub-design matrix isbased on a modelled contribution to the fingerprint from the respectivedevice and substrate.
 9. The method of claim 2, wherein the equationcomprising the matrix is solved using: i. least squares fit, optionallyusing regularization, and further optionally using an L-curve methodand/or leave-one-out cross validation; and/or ii. Bayesian statistics.10. The method of claim 1, further comprising analyzing variation of theparameter data using the usage data, and wherein determining thecontribution to the parameter for a device comprises grouping the datausing the analyzed variation.
 11. The method of claim 1, wherein atleast one of the devices is controlled based on the determinedcontribution to the fingerprint of the parameter of that device.
 12. Themethod of claim 1, wherein the parameter is one or more selected from:i. critical dimension, overlay, critical dimension uniformity, line edgeplacement, alignment, focus, pattern shift, line edge roughness, microtopology, and/or edge placement error; and/or ii. a shape description ofa feature, such as side-wall angle, resist height, and/or contact holeellipticity; and/or iii. a processing parameter such as a coatingthickness, optionally bottom anti-reflective coating thickness and/orresist thickness, and/or an optical property of a coating which mayoptionally indicate a measure of absorption, such as refractive indexand/or extinction coefficient; and/or iv. a parameter determined fromsubstrate measurements, such as a yield parameter, optionally defectand/or electrical performance.
 13. The method of claim 1, wherein theparameter data is based on processed measurements, and wherein themeasurements are processed using principal component analysis or fittingof a model, desirably a polynomial model or a linear model.
 14. A systemcomprising a processor configured to determine a contribution of adevice out of a plurality of devices to a fingerprint of a parameter,the parameter being associated with processing of a substrate, theprocessor configured to at least: obtain parameter data and usage data,wherein the parameter data is based on measurements for multiplesubstrates having been processed by the plurality of devices, and theusage data indicates which of the devices out of the plurality ofdevices were used in the processing of each substrate; and determine thecontribution using the usage data and the parameter data.
 15. A programto control determining a contribution of a device out of a plurality ofdevices to a fingerprint of a parameter, the parameter being associatedwith processing of a substrate, the program comprising instructions forcarrying out at least: obtaining parameter data and usage data, whereinthe parameter data is based on measurements for multiple substrateshaving been processed by the plurality of devices, and the usage dataindicates which of the devices out of the plurality of devices were usedin the processing of each substrate; and determining the contributionusing the usage data and the parameter data.